Phase-driven charge manipulation in Hybrid Single-Electron Transistor

Phase-tunable hybrid devices, built upon nanostructures combining normal metal and superconductors, have been the subject of intense studies due to their numerous combinations of different charge and heat transport configurations. They exhibit solid applications in quantum metrology and coherent caloritronics. Here we propose and realize a new kind of hybrid device with potential application in single charge manipulation and quantized current generation. We show that by tuning superconductivity on two proximized nanowires, coupled via a Coulombic normal-metal island, we are able to control its charge state configuration. This device supports a one-control-parameter cycle being actuated by the sole magnetic flux. In a voltage biased regime, the phase-tunable superconducting gaps can act as energy barriers for charge quanta leading to an additional degree of freedom in single electronics. The resulting configuration is fully electrostatic and the current across the device is governed by the quasiparticle populations in the source and drain leads. Notably, the proposed device can be realized using standard nanotechniques opening the possibility to a straightforward coupling with the nowadays well developed superconducting electronics.

The manipulation of single charge quanta in solid state systems has been a field of active interest for several decades [1,2]. At the basis of such structures lays the Coulombic repulsion between electrons in a small metallic island yielding the so-called Coulomb-blockade regime [3]. Many quantum phenomena, such as the tunneling effect [4] or the Josephson coupling [5] in superconducting junctions, have been exploited so far to control the charging state of a conducting island, and enable a number of applications ranging from on-chip cooling [6][7][8][9] and current generation [10][11][12] to single-photon detection [13]. Yet, in all Coulombic systems the external control parameters play a crucial role, reaching sometimes a rather complex experimental configuration [14]. Here we show that quantum interference established in a superconducting nanowire [15] can provide a phase-tunable energy barrier which allows charge manipulation with enhanced functionalities [16]. This additional degree of freedom for single electronics stems from the coupling between magnetic flux-dependent proximity effect [17][18][19] and charging levels discretization present in a Coulombblockaded island. Our quantum interference nanostructure represents the first realization of a superconducting hybrid single-electron transistor called SQUISET [16] in which the charging landscape is phase-coherently manipulated by an external magnetic flux. The interferometric nature of the transistor adds new perspectives to single electronics making it a promising alternative building block in quantum metrology [20], coherent caloritronics [17,21,22], and quantum information technology.
When a short metallic nanowire weak-link interrupts a superconducting ring its electronic density of states (DOSs) can be strongly modified by an externally-applied magnetic flux piercing the loop [23]. This property is at the basis of the superconducting quantum interference proximity transistor (SQUIPT). The latter exploits a flux-tunable DOSs to achieve high sensitivity for detecting magnetic fields. So far, SQUIPTs have been implemented in several configurations [15,18,24,25]. In this context, the behavior of a fully-superconducting SQUIPT, i.e., where the nanowire itself is a superconductor S', has been recently theoretically investigated [23]. In particular, it was shown that if the S' nanowire lenght (L) is smaller or comparable to the superconducting coherence length its DOSs shows a flux-tunable BCS-like temperature-dependent (T ) energy gap, ∆ (T, Φ) = ∆ (T ) |cos (πΦ/Φ 0 )| where Φ the magnetic flux piercing the loop, and Φ 0 2 × 10 −15 Wb is the flux quantum. The above functional behavior for the gap has been recently pro-posed as the building block of a Coulombic turnstile called superconducting quantum interference single-electron transistor (SQUISET) [16]. In the SQUISET setup, the charging state of a metallic island in the Coulomb blockade regime can be controlled by manipulating the DOSs of its source and drain SQUIPT electrodes via a suitable applied magnetic field.
Here we report the first realization of a SQUISET and its phase-coherent magneto-electric behavior. For the experiment we adopted two SQUIPTs based on S' weak-links. Our design, exploiting short superconducing nanowires, guarantees the presence of a well-defined energy gap in every measurement configuration.
The SQUISETs are fabricated by standard three-angle shadow-mask [26] deposition of metals through a conventional suspended resist mask (see Methods). A pseudo-color scanning electron micrograph of the core of one typical device is shown in Fig. 1a. Two 150-nm-thick superconducting aluminum (Al) rings having different areas A 1 and A 2 are interrupted by two Al nanowires with length L ≈ 150 nm, thickness t ≈ 20 nm and width w ≈ 35 nm. The energy gap in both S' nanowires is ∆ 0 = ∆ (0) ≈ 198 µeV, and they realize the source and the drain electrodes of our transistor. The two SQUISET loops are pierced by a uniform magnetic field leading to different Φ 1 and Φ 2 magnetic fluxes. In addition, the superconducting nanowires are galvanically connected each other through a normal metal island capacitively-coupled to a gate electrode, the couplings being tunnel junctions with normal-state resistance R T ≈ 63 kΩ. In the experiment the SQUISET is voltage biased across source and drain electrodes (V SD ), and the gate capacitor is polarized with n G = C G V G /e elementary charges, where V G is gate voltage, C G its capacitance, and e is the electron charge. Figures 1b-d represent a typical clocking sequence where a single electron is trasnferred from drain to source by proper modulation of the two DOSs in a fully electrostatic configuration. The source-drain current flowing through the SQUISET can be written as (see Methods for details) [16] where E i,± = ±2E c (n − n G ± 1/2) ± eV SD /2 are the free energy variations associated to a single electron tunneling event in the i-th junction that increase (+) or decrease (−) the number of excess charges on the island (n), and Γ 1,± (Φ 1 , E 1,± ) are the tunneling rates across the first junction. Moreover, is the probability to find the transistor in the n charging state depending on its electro-magnetic environment. The full expressions for the above quantities can be The island (realized in Al 0.98 Mn 0.02 , yellow) is tunnel-coupled through identical junctions of normal-state resistance R T ≈ 63 kΩ to both weak-links, and is capacitively-coupled to the gate electrode (light blue). b-d, Energy band diagrams which captures the physics governing electron transport through the SQUISET. In (b) a single electron is loaded into the island from the drain electrode. In (c) the previous electron is blocked onto the island. In (d) the island is discharged and, as a consequence of the entire process, one single electron is transferred from the drain electrode to the source.
found in [16], and obey the standard "orthodox theory" of single-electron tunneling [3]. For small magnetic fields, i.e., for B B c where B c 10.5 mT is the Al critical magnetic field, the tunneling rates across the i-th junction of the SQUISET follow the model described in [16] where the magnetic flux associated to the i-th loop (Φ i ) tunes the DOSs of the i-th S' nanowire. The probability p n can then be found recursively from the calculation of the tunneling rates [1].
The bare charging energy contribution to electron transport in the SQUISET can be determined in high magnetic fields, i.e., at 50 mT. In this condition, superconductivity in the nanostructure is suppressed so that the current I SD become magnetic-flux independent. From the transistor stability diagram in the normal state [see Fig.2d] we deduce a charging energy of Ec 24 µeV.
Yet, the structure behavior in the superconducting state is drastically different. In particular, Figs. 2a-c reveal that by piercing the SQUISET with a uniform perpendicular magnetic field smaller than B c yields a beating-like dependence of I SD on the magnetic flux Φ B . Here, we conveniently introduced a beating magnetic flux, Φ B = (Φ 1 − Φ 2 )/2, whose role is clearly visible in the envelope functions displayed in Fig. 2c. Furthermore, a frequency analysis of the current flowing through the transistor allows to deduce the loops areas of A 1 5 µm 2 and A 2 6.5 µm 2 (see Methods). With the aid of the above quantities we can easily describe the flux-dependent behavior of the energy gaps in the two S' nanowires [see extending this latter concept, we acquired an alternative stability diagram fixing the gate voltage to n G = 0.5 and varying Φ B [see Fig. 2d]. Here, current beatings are clearly visible in the whole spectrum under investigation, and are uniquely related to the flux-dependent superconducting gaps in the two S' nanowires having dropped out the Coulomb-blockade contribution with a proper biasing of the gate electrode.
Having quantified the role of parameters n G and Φ B , we proceed now to investigate the regimes where these are coupled. Charging effects are usually presented in the form of Coulomb oscillations [see Fig. 3 and in particular Fig. 3e], where a fixed source-drain voltage (V SD ) selects a suitable energy window for which the current through the transistor is modulated by the gate voltage. In the SQUISET, this energy window can be shifted [ Fig. 3e] thanks to flux-tunable superconducting energy gaps under their beating-like behavior as a function of Φ B . Figures 3a-d display the contour plots of Coulomb oscillations vs Φ B and n G for four selected source-drain voltage values (V SD = 425µV, V SD = 325µV, V SD = 225µV and V SD = 125µV from top to bottom, respectively). For large values of V SD [see Fig. 3a], the source-drain current is allowed to flow for the whole parameters space apart from small lobes (two-dimensional analogue of the Coulomb peaks) located near Φ B = 0 and Φ B = Φ 0 , with a unitary periodicity in the n G axis. In these conditions the two superconducting energy gap are almost in phase, and at their maximum. Their energy contribution can be deduced from our model [see Fig. 2a-b] yielding ∆ 0 198 µeV. In this condition the gate voltage modulates the current from a blockaded (brown) to a conducting region (orange) by exploiting the charging energy effect.
For lower bias voltage [see Fig. 3b-c] extended plateaus of zero current appear around Φ B ∼ 0 and Φ B ∼ Φ 0 due to a reduced available energy which is not sufficient to transfer any electron across the tunneling barriers. Accordingly, the Coulomb lobes appear shifted in magnetic flux reaching the regions of Φ B = 0.5Φ 0 and Φ B = 1.5Φ 0 for V SD = 225 µeV. There, the energy gap functions are out of phase [see  Fig. 2a-b] and their effect is then limited in energy (∆ 1 + ∆ 2 < eV SD = 225 µeV). In Fig. 3d one can observe how Coulomb oscillations are still present around Φ B = 1.0, where the two gaps are in phase at their minimum. In such a low bias regime (V SD = 125 µV) the current flow is still modulated by the gate, revealing an almost complete closure of both the two energy gaps (∆ 1 + ∆ 2 < eV SD = 125 µeV). This latter pheomenon is evident in the transconductance characteristics shown in Fig.  3f where, selected a condition (Φ B = 1.33Φ 0 ) for which ∆ 1 and ∆ 2 are almost at their minimum, the gate control parameter effect is maximum around integer values of n G and minimum for n G half integer values. As a confirmation of the whole presented model, the smaller is V SD there, the more prounounced is the gate effect.
We now show the turnstile-like behavior of our singleelectron transistor in the full out-of-phase regime (around Φ B = 0.5Φ 0 ). In Fig. 4 [16] is effectively realized, and allows us to investigate the modulation of the island charging state with the sole action of the magnetic flux. Figures 4a-c represent the energy band diagrams using the real electromagnetic and energetic transistor configurations depicted by the green,red and blue circles in Fig 4g-i. There it clearly appears how the island discrete energy levels arising from the Coulomb blockade are filled and discharged following the full and empty branches of the nanwires DOSs. The magnetically-gated turnstile [27] configuration is confirmed by the twisting-like evolution of the measured stability diagrams [see Fig. 4d-f], and it's in good agreement with the theoretical model [shown in Fig. 4g-i]. In the latter the charging landscape is represented by Coulomb diamonds with different colors following the p n evolution vs. Φ B ; pure RGB colors correspond to a blockaded FIG. 4. SQUISET working principle: comparison between experiment and theory a-c, Energy band diagrams of three representative electro-magnetic configurations of the transistor. Here we set n g = 0 and eV SD = ∆ 0 /2. It clearly appears the influence of the magnetic flux on the number of excess charges on the island being in the blockade regime and at fixed bias voltage. The empty and the full branches of the quasiparticles DOSs are vertically shifted by the external magnetic flux loading or unloading different charging levels (n). d-f, Color-plot showing the measured source-drain differential conductance (G = ∂ I SD ∂V SD ) stability diagrams. g-i, Corresponding calculated stability diagrams. Red channel is proportional to p −3 + p 0 + p 3 , green channel is proportional to p −2 + p 1 + p 4 , and blue channel is proportional to p −4 + p −1 + p 2 . Black dots indicate the configurations depicted in the diagrams of panels (a-c). Plots (a), (d) and (g) were obtained for Φ B = 0.359Φ 0 . In (b), (e) and (h) we set Φ B = 0.393Φ 0 , whereas in (c), (f) and (i) is shown the condition for Φ B = 0.5Φ 0 . All measurements were taken at 21 mK of bath temperature. state where the number of occupied energy levels (n) is fixed in time. Corresponding energy levels are filled in Fig. 4ac with coloured circles representing the electrons participating in the quantized charging state of the island. Eventually, Figure 4 represents an example flux sequence, starting from Φ B = 0.359Φ 0 reaching Φ B = 0.5Φ 0 , which demonstrates the possibility to control the transfering of two electrons [red and green in Fig. 4c] from drain to source electrodes in a fully electrostatic regime.
We conclude by emphasizing the role of the flux control parameter: our experimental results show indeed how, with a proper magnetic-like gating, a SQUISET having a rather simple design can act as a turnstile for single charge quanta.
In summary, we have realized the first superconducting quantum interference single-electron transistor where the magnetic flux degree of freedom provides a highly-efficient charge state quantization. Being based on the series of two SQUIPTs coupled to a gate electrode, the transistor can be easily realized with conventional nanofabrication techniques. In addition, it can potentially be combined with on chip coils or back-action feedback loops for enhanced control. The fully-quantum nature intrinsic to the flux-tunable superconducting gap governing the SQUISET enables phase-coherent single-electron circuitry to be investigated. Eventually, our transistor concept might have impact in cryogenic microelectronics as well as in solid-state quantum information technologies. Yet, combined with superconducting resonators the SQUISET could enable innovative realizations as a building block of future coherent single nanoelectronics.

METHODS
Experimental -One step of electron-beam lithography was used to fabricate the transistors followed by a three-angle shadow-mask evaporation of metallic thin films through a suspended resist mask. The latter, lying onto an oxidized Si substrate, was processed in a UHV electron-beam evaporator. A 15-nm-thick layer of Al 0.98 Mn 0.02 was initially deposited, the chip being tilted at an angle of 37 • , to realize the normal metal island structure. The sample was then exposed to 4 × 10 −2 mbar O 2 for 5 min to form the AlO x thin insulating layer composing the tunnel junctions. Then the chip was tilted to 20 • , and a 25-nm-thick layer of Al was deposited to form the superconducting (S') nanowires. Finally, a 150-nm-thick layer of Al was deposited at 0 • to realize the superconducting rings. The source and the drain electrodes are nominally identical differing only for the rings areas.
The magneto-electric characterization of the SQUISETs was performed down to 20mK in a shielded and filtered dilution refrigerator. Current and conductance measurements were performed with room temperature electronics whereas the magnetic field was imposed through a superconducting coil surrounding the sample chamber.
Theory -The theoretical modeling of the SQUISET was performed in the framework of the "orthodox theory' of single-electron tunneling under the static regime condition. The sequential tunneling approach in the first order approximation leads to Eq. 1 as a solution of the standard master equation [1]. We assumed each component of the transistor to reside at thermal equilibrium with the lattice phonons at the base temperature of the criostat. Intragap quasiparticles populations in the source and drain electrodes have been modeled by assuming a smeared superconducting density of states with a realistic Γ 0 = 10 −5 ∆ 0 Dynes parameter [28,29]. The magnetic fluxes, Φ i = A i B, have been deduced assuming a uniform magnetic field B perpendicular to the SQUIPTs loop plane, and areas (A i ) coming from the experimental evaluation. Moreover, the two areas were also estimated by peak analysis on the FFT trace of the current vs magnetic field signal, the latter revealing two primary components related to the two different pierced loops.